A free boundary problem with nonlocal diffusion and unbounded initial range

A free boundary problem with nonlocal diffusion and unbounded initial range | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A free boundary problem with nonlocal diffusion and unbounded initial range Lei Li, Xueping Li, Mingxin Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1762946/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract We consider a free boundary problem with nonlocal diffusion and unbounded initial range, which can be used to model the propagation phenomenon of an invasion species whose habitat is the interval $(-\infty,h(t))$ with $h(t)$ representing the spreading front. Since the spatial scale is unbounded, a different method from the existing works about nonlocal diffusion problem with free boundary is employed to obtain the well-posdeness. Then we prove that the species always spreads successfully, which is very different from the free boundary problem with bounded range. We also show that there is a finite spreading speed if and only if a threshold condition is satisfied by the kernel function. Moreover, the rate of accelerated spreading and accurate estimates on longtime behaviors of solution are derived. AMS Subject Classification (2000): 35K57, 35R09, 35R20, 35R35, 92D25 Nonlocal diffusion free boundary unbounded range accelerated spreading Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Major revision 16 Jul, 2022 Reviews received at journal 06 Jul, 2022 Reviewers agreed at journal 25 Jun, 2022 Reviewers invited by journal 25 Jun, 2022 Editor assigned by journal 18 Jun, 2022 Submission checks completed at journal 18 Jun, 2022 First submitted to journal 15 Jun, 2022 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-1762946","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":114486702,"identity":"36df8b52-af03-4c0d-abec-c0fcb6b00cea","order_by":0,"name":"Lei Li","email":"","orcid":"","institution":"Henan University of Technology","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Lei","middleName":"","lastName":"Li","suffix":""},{"id":114486703,"identity":"4a7ce006-fd5b-43b5-abb7-69893a14952f","order_by":1,"name":"Xueping Li","email":"","orcid":"","institution":"Zhengzhou University of Light Industry","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"Xueping","middleName":"","lastName":"Li","suffix":""},{"id":114486704,"identity":"b2a89a9d-604f-4717-926e-f3d3a3963bf7","order_by":2,"name":"Mingxin Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAwklEQVRIiWNgGAWjYPACmwQwxUOCljTStRwmQYvujBzDzwW/zueZSyQwPnjbxiBvTkiL2ZkzxtIz+24XW85IYDac28ZguLOBkJbjvRukeXtuJ264kcAmzdvGkGBwgJCWw7ybf/P2nANpYf9NnJbjvdukeX4cANvCTJyWM+e/WfM2JCfu7HnYLDnnnIThBoJabqQl3+b5Y5e4nT354Ic3ZTbyBG0BA8Y2BgYDBsYGIFOCGPUg8AekZRSMglEwCkYBDgAAYOhDwYeN0YQAAAAASUVORK5CYII=","orcid":"","institution":"Henan Polytechnic University","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"Mingxin","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2022-06-16 01:29:14","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1762946/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1762946/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":23007745,"identity":"f6de6b1d-d1f7-4702-889a-0efb99adb6ec","added_by":"auto","created_at":"2022-06-23 16:07:35","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":303555,"visible":true,"origin":"","legend":"","description":"","filename":"LLWfreebunboundedZAMP.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1762946/v1_covered.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A free boundary problem with nonlocal diffusion and unbounded initial range","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1762946/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"zeitschrift-fur-angewandte-mathematik-und-physik","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"zamp","sideBox":"Learn more about [Zeitschrift für angewandte Mathematik und Physik](http://link.springer.com/journal/33)","snPcode":"33","submissionUrl":"https://submission.nature.com/new-submission/33/3","title":"Zeitschrift für angewandte Mathematik und Physik","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Nonlocal diffusion, free boundary, unbounded range, accelerated spreading","lastPublishedDoi":"10.21203/rs.3.rs-1762946/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1762946/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe consider a free boundary problem with nonlocal diffusion and unbounded initial range, which can be used to model the propagation phenomenon of an invasion species whose habitat is the interval $(-\\infty,h(t))$ with $h(t)$ representing the spreading front. Since the spatial scale is unbounded, a different method from the existing works about nonlocal diffusion problem with free boundary is employed to obtain the well-posdeness. Then we prove that the species always spreads successfully, which is very different from the free boundary problem with bounded range. We also show that there is a finite spreading speed if and only if a threshold condition is satisfied by the kernel function. 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